Bench Measurements and Simulations of Beam Coupling Impedance
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1 Bench Measurements and Simulations of Beam Coupling Impedance Uwe Niedermayer Institut für Theorie Elektromagnetischer Felder, TU-Darmstadt, Germany CERN Accelerator School on Intensity Limitations in Particle Beams 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 1
2 Maxwell s Equations and Particle Beams Maxwell Interpolate Not suitable for complicated PIC accelerator structures Lorentz Force Self consistent for Update 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 2
3 Wake fields and Beam Coupling Impedance: Divide and Conquer Maxwell Electromagnetics Mechanics Wake function or impedance Update 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 3
4 Maxwell s equations Pictures: Wikipedia Transformation by laws of Gauss and Stokes J.C. Maxwell J. W. Gibbs Frequency Domain: Material relations: (macroscopic) 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 4
5 Contents Intro: wake functions and beam coupling impedance Electromagnetic field simulations Examples with the Finite Integration Technique (FIT) Overview of simulation techniques Frequency domain simulations Wire bench measurements Primer on S-parameters and Vector Network Analysis (VNA) Connection between S-parameters and beam coupling impedance Modeling and simulation of the wire measurement TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 5
6 Wake Function Test charge Source charge Arbitrary accelerator structure Characterizes the accelerator structure Kick after passage Green s function Wake potential by convolution Beam Coupling Impedance by Fourier Transform Picture: Consequences on Beam Dynamics Can lead to beam instabilities (both longitudinal and transverse) Describes beam induced component heating 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 6
7 Contents Intro: wake functions and beam coupling impedance Electromagnetic field simulations Examples with the Finite Integration Technique (FIT) Overview of simulation techniques Frequency domain simulations Wire bench measurements Primer on S-parameters and Vector Network Analysis (VNA) Connection between S-parameters and beam coupling impedance Modeling and simulation of the wire measurement TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 8
8 Mesh In general Mesh and Method are independent. Usually FEM on tetrahedral mesh (unstructured) Usually FIT/FDTD on hexahedral mesh (structured) Pictures: Wikipedia Computational advantages Modeling advantages 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 9
9 Finite Integration Technique (FIT): Grid-Maxwell-Equations The Grid-Equations represent an EVALUATION of Maxwell s equations Therefore they are exact Lecture: Verfahren und Anwendungen der Feldsimulation, TUD 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 10
10 FIT Topological relations Dual edge Primal face Topological curl (C) Topological div (S) Mesh duality Dual node Primal node Dual orthogonal mesh leads to diagonal material matrices 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 11
11 Leapfrog algorithm and stability Courant Friedrichs Levy (CFL) stability criterion: For equidistant mesh and vacuum Moreover: grid dispersion, numerical Cherenkov radiation, 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 13
12 Wake Potential Example (Broadband) Ferrite Ring in Perfectly Electric Conducting (PEC) Pipe 10 cm Magnitude of the electric field, logarithmic color scale Longitudinal cut CST Particle Studio Beam line density (a.u.) Wake potential (convolution of wake function with line density) Dispersively lossy ferrite material 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 14
13 Impedance of the ferrite ring (normalized DFT of wake potential) Ferrite Amidon Material 43 Fitting of material data on impulse response model required! 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 15
14 Wake Potential Example (Narrowband) Parasitic cavity with 2 gaps (arbitrary ) Rotationally symmetric 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 16
15 Longitudinal impedance (unfiltered and coarse mesh) 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 17
16 Introducing a Ferrite Ring to damp the modes in the structure Rotationally symmetric 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 18
17 Shifting the ring 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 19
18 Increasing the length of the ring 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 20
19 Magnetic field Rotationally symmetric So we have to look at the magnetic field of the Eigenmodes of the structure 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 21
20 First Mode H-Field Rotationally symmetric 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 22
21 Second Mode H-Field Rotationally symmetric 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 23
22 Contents Intro: wake functions and beam coupling impedance Electromagnetic field simulations Examples with the Finite Integration Technique (FIT) Overview of simulation techniques Frequency domain simulations Wire bench measurements Primer on S-parameters and Vector Network Analysis (VNA) Connection between S-parameters and beam coupling impedance Modeling and simulation of the wire measurement TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 24
23 Overview of methods and codes Explicit TD: CST-PS, GdfidL, Echo, ABCI, extremely fast Implicit TD: ACE3P (SLAC), not limited by CFL, computationally expensive FD: BeamImpedance2D, nice for 2D (x,y) problems Using commercial software tools without beam (e.g. HFSS) for the determination of beam coupling impedance See e.g. - T. Kroyer, CERN AB-Note Niedermayer and Boine-Frankenheim NIM A Kononenko and Grudiev PRSTAB November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 25
24 Impedance of SIS100 dipole chamber (FD power loss calculation with CST-EMS) Longitudinal E-field and wall current for Dipolar wall current Lowest relevant frequency (specified by Betatron tune) Neclect of beam charge valid at LF and high beam velocity U.Niedermayer and O. Boine-Frankenheim, Analytical and numerical calculations of resistive wall impedances for thin beam pipe structures at low frequencies, NIM A, November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 26
25 Properties of Time Domain (TD) and Frequency Domain (FD) Computation (Explicit-) TD (Impedances obtained by DFT) Broadband Matrix-vector products (cheap) Commercial / non-commercial codes available FD (Impedances obtained directly) Arbitrary frequency points Beam velocity and dispersive material data are just parameters Limitation at Low Frequency by uncertainty relation and CFL criterion Difficult for low beam velocity Difficult for dispersive material Computationally expensive (one ill conditioned linear system of equations (LSE) for each frequency) Optimized codes not (yet) available 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 27
26 Contents Intro: wake functions and beam coupling impedance Electromagnetic field simulations Examples with the Finite Integration Technique (FIT) Overview of simulation techniques Frequency domain simulations Wire bench measurements Primer on S-parameters and Vector Network Analysis (VNA) Connection between S-parameters and beam coupling impedance Modeling and simulation of the wire measurement TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 28
27 Defining Sources for Impedance Computation in the Frequency Domain charged disc displaced source = undisplaced source + dipole moment at the boundary Monopole moment Dipole moment Longitudinal wake function / impedance Transverse wake function / impedance 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 29
28 Beam Coupling Impedance Fourier transform of wake function Fourier transform Longitudinal monopolar impedance Transverse dipolar impedance Volume integral definition well suited for computation on mesh 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 30
29 Frequency Domain Computation: Curl-Curl Equation and FIT FIT Complex linear system of equations (LSE) size 3N p Dedicated boundary conditions required for the entry and exit of the beam Longitudinal Phaseshift given a priori Floquet (periodic) boundary conditions: Connection of the two boundaries with proper phase advance U.Niedermayer and O. Boine Frankenheim, Proc. of ICAP 2012 and references therein 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 31
30 Frequency Domain Computation: Curl-Curl Equation and FIT FIT Complex linear system of equations (LSE) size 3N p Dedicated boundary conditions required for the entry and exit of the beam Longitudinal Phaseshift given a priori Floquet (periodic) boundary conditions: U.Niedermayer and O. Boine Frankenheim, Proc. of ICAP 2012 and references therein 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 32
31 Dipole Source for Transverse Impedance z y x Dipole: two beams with opposite charge Dipole: ring shaped beam Direct space charge fields cannot be properly modeled! Direct fields are known analytically and can be subtracted! 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 33
32 Finite Element Method (FEM) Very flexible due to unstructured mesh Discretization of weak formulation Standard Ritz-Galerkin FEM: trial and test functions are identical 2D impedance solver implemented Open source package FEniCS (A. Logg, K. Mardal, G. Wells et al.) Mesh from Gmsh (C. Geuzaine, J. Remacle) all open source Weak formulation of PDE can be interpreted No complex numbers, can be overcome by coupled function spaces 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 34
33 Discretization of the Electromagnetic Problem in 2D Split: Longitudinal / Transverse Real / Imaginary Discretized by 1 st order nodal elements Discretized by 1 st order Nédélec edge elements of the first kind Pic.: P. Jacobsson, Nedelec elements for computational electromagnetics 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 35
34 Helmholtz Split (Domain needs to be simply connected) solenoidal irrotational Continuity equation 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 36
35 Ritz-Galerkin FEM Discretization Few (<10 7 ) degrees of freedom (dof) in 2D solved with sparse direct solver U. Niedermayer et al., Space charge and resistive wall impedance computation in the frequency domain using the finite element method, Phys. Rev. STAB 18, , 2015 Code named BeamImpedance2D (in PYTHON) 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 39
36 Lonitudinal Impedance Examples Longitudinal space charge impedance A ferrite ring (dispersively lossy material) Beam of radius a=1cm in perfecly conducting pipe of radius b=4cm Asymptotes: 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 40
37 Application: Beam Induced Heat Power in SIS 100 transfer kicker magnet Magnet gap Dispersive ferrite yoke Angular revolution frequency Beam power spectral density Beam Beam spectrum envelope (a.u.) 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 41
38 Transverse Impedance Example Transverse space charge impedance indirect Picture: A. W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators Full (Direct+Indirect) transverse space charge impedance direct Source representation on FEM mesh (a.u.) Indirect transverse space charge impedance 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 42
39 Transverse Resistive Wall Impedance: Thin Steel Beam Pipe (idealized SIS-100 pipe) Thin wall Thick wall Surface impedance boundary condition Resolved wall d=0.3mm 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 43
40 Contents Intro: wake functions and beam coupling impedance Electromagnetic field simulations Examples with the Finite Integration Technique (FIT) Overview of simulation techniques Frequency domain simulations Wire bench measurements Primer on S-parameters and Vector Network Analysis (VNA) Connection between S-parameters and beam coupling impedance Modeling and simulation of the wire measurement TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 44
41 Motivation for Wire Bench Measurements: Source Fields Coulomb field of point charge x Observation point Lorentz transformation to lab-frame y v z Source Fields 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 45 TEM Mode!
42 Motivation and History of Wire Bench Measurements Beam Bench M. Sands, J. Rees, SLAC Report PEP-95, 1974 The two setups produce approximately the same image current in the wall Measurements of loss factors in time domain 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 46
43 It has evolved since 1974 Read as many papers as possible before you start! - F. Caspers, article in the Handbook of Accelerator Physics and Engineering - F. Caspers and A. Mostacci, talk given at ICFA mini workshop on wakefields and impedance, Erice G. Nassibian and F. Sacherer, Methods for Measuring Transverse Coupling Impedances, Nucl. Instrum. Meth., vol. 159, no. 6, pp , H. Hahn and F. Pedersen, On Coaxial Wire Measurements of the Longitudinal Coupling Impedance, BNL Report 78-9, T. Kroyer, F. Caspers, and E. Gaxiola, Longitudinal and Transverse Wire Measurements for the Evaluation of Impedance Reduction Measures on the MKE Extraction Kickers, CERN Rep., V. Vaccaro, Coupling Impedance Measurements: An improved wire method, INFN/TC-94/023, E. Jensen, An improved log-formula for homogeneously distributed impedance, PS/RF/ A. Argan, L. Palumbo, M. R. Masullo, and V. G. Vaccaro, On the Sands and Rees Measurement Method of the Longitudinal Coupling Impedance, Proc. of PAC 8, F. Caspers, C. Gonzalez, M. D yachkov, E. Shaposhnikova, H. Tsutsui, Impedance Measurement Of The SPS MKE Kicker By Means Of The Coaxial Wire Method, PS/RF/Note E. Métral, F. Caspers, M. Giovannozzi, A. Grudiev, T. Kroyer, and L. Sermeus, Kicker impedance measurements for the future multiturn extraction of the CERN Proton Synchrotron, in EPAC, Edinburgh, Many more, see proceedings 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 47
44 Contents Intro: wake functions and beam coupling impedance Electromagnetic field simulations Examples with the Finite Integration Technique (FIT) Overview of simulation techniques Frequency domain simulations Wire bench measurements Primer on S-parameters and Vector Network Analysis (VNA) Connection between S-parameters and beam coupling impedance Modeling and simulation of the wire measurement TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 48
45 Wave Amplitudes and Scattering Parameters Voltage cannot be uniquely defined in RF systems Thus one defines power flow parameters a i and b i (Unit: sqrt(w)) which are related to the voltage and current in a TEM line by Smith Chart maps between z and 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 49
46 Scattering parameters measurement For simplicity we consider only TEM modes with same Z c at each port Vector Network Analyzer (VNA) Examples of calibration standards Pics: co.uk/support/85033/hp/ Measures S-parameters vector means mag and phase Stimulus and detection Calibration of cables to reference plane required (SOLT) Phase stable coaxial cables required Can convert S,Z,Y,T paramerters internally Can display Bode, Smith, Polar 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 50
47 Internal Setup of the VNA Most VNA work with heterodyne detection (i.e. 2 oscillators) IF filter (digital FIR filter) bandwidth to be chosen trade-off between noise level and filter filling time (sweep time) Averaging 4-port VNAs offer conversion to symmetric S-parameters Picture: M. Hiebel, Fundamentals of Vector Network Analysis, published by Rohde&Schwarz, November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 51
48 Example: Calculation of S-parameters for a resistor within a transmission line Lumped element 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 52
49 Contents Intro: wake functions and beam coupling impedance Electromagnetic field simulations Examples with the Finite Integration Technique (FIT) Overview of simulation techniques Frequency domain simulations Wire bench measurements Primer on S-parameters and Vector Network Analysis (VNA) Connection between S-parameters and beam coupling impedance Modeling and simulation of the wire measurement TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 53
50 Bench Measurements of Broadband Impedances Measurement in the Frequency Domain VNA Measure transmission parameter S 21 Two fundamental issues - Corresponds only to scaling by FD simulation - Wire must be thin high characteristic impedance DUT REF 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 54
51 A Priori Distinction between Lumped and Distributed Impedance De-embedded structure: DUT Characteristic Impedance Lumped (concentrated) impedance Equally distributed impedance Hahn and Pedersen November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 55
52 Distributed Impedance cont d REF DUT Textbook, e.g. Pozar Improved Log Formula, Vaccaro et al DUT J.Wang and S.Zhang,, NIM A 459, November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 56
53 What if the impedance is neither lumped nor equally distributed? Mixed Impedance: Log-Formula Log-Formula (Walling et al. 1989) Requirements for lumped impedance: Requirements for distributed impedance: electrical length U. Niedermayer et al., Analytic modeling, simulation and interpretation of broadband beam coupling impedance bench measurements, Nucl. Instrum. Meth. A 776, 2015 H. Hahn, Validity of coupling impedance bench measurements, PRSTAB 3, , November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 57
54 Benchmarking the different formulas in 2D (analytical) (slow) convergence for wire radius Beam impedance calculated analytically (multilayer field matching) S 21 calculated from Eigenvalue equation for k z of Quasi-TEM mode (semi-analytically) U. Niedermayer et al., Analytic modeling, simulation and interpretation of broadband beam coupling impedance bench measurements, Nucl. Instrum. Meth. A 776, November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 58
55 Benchmarking the different formulas in 3D (numerical) Without reflection correction With reflection correction Reflection correction Easy in simulation waveguide ports Difficult in reality! Multiple reflections Beam impedance from CST Particle Studio (PS) S-parameters from CST Microwave Studio (MWS) 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 59
56 Measurement of the ferrite ring Attenuation foam 3.3 cm measurement uncertainty Large setup Small setup L. Eidam 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 60
57 Measurement of Transverse Impedance Twin wire measurement S 21 Z formulas can be linearized Lumped and distributed impedance do not have to be distinguished Expectation by microwave simulation: L. Eidam Upper frequency limit due to coil approx. 3 MHz Reasonable measurement results down to 1 khz Quasi-stationary interpretation required 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 61
58 Low frequency measurement of beam pipe transverse impedance Analytical L. Eidam Upper frequency limit due to coil approx. 3 MHz Reasonable measurement results down to 1 khz Quasi-stationary interpretation required 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 62
59 Summary Frequency domain beam coupling impedance computation Advantageous for LF, dispersive material, low beam velocity Direct transverse space charge fields not accurate on structured hex-mesh Space charge and resistive wall impedance solver in 2D FEM implemented SIBC allows high frequency since skin depth is not meshed Bench measurements of broadband impedances Should be cross-checked with RF simulations A priori knowledge about impedance distribution required for Medium frequency range Twin wire method inapplicable at LF by coil method down to 1 khz Quasi stationary methods at extremely LF, for both simulation and measurement 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 63
60 Outlook I will be available the whole week for any questions Optimized (parallel) FEM-FD solver in 3D (PhD position open at TUD) Optimization of matching networks and conducting bench measurements (also above cut-off) (BSc/MSc projects open at GSI and TUD) 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 64
61 The End Thank you for your attention! Any Questions? Please contact me for more references Acknowledgements: - Elias Metral and ICE section at CERN - Fritz Caspers, Manfred Wendt (CERN) - Lewin Eidam, Udo Blell, Oliver Boine-Frankenheim (GSI) - Wolfgang Ackermann, Erion Gjonaj, Ulrich Römer, Herbert De Gersem, Thomas Weiland and many more (TEMF) 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 65
62 Instability Example: Transverse Coasting Beam Instability V. Kornilov, GSI Exponential growth Coherent beam oscillation Ideal beam trajectory Betatron frequency corresponds to a very low frequency ~100 khz Slice of beam pipe Magnitude of the longitudinal E-field (dipole excitation) Details on the low frequency (LF) impedance computation: U. Niedermayer and O. Boine-Frankenheim, Analytical and numerical calculations of resistive wall impedances for thin beam pipe structures at low frequencies, Nucl. Instrum. Meth. A 687, 51, November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 66
63 De Rham Diagram and Discretization of Sobolev Spaces De Rham sequence Hodge operators (invertible) Supplement / Restrict Vectorial and scalar 2D curl operators Similar De Rham diagram for discrete spaces Functions can be projected from continous to discrete If the projecton operator commutes with the exterior derivative, then the convergence of the projection implies the convergence of the FEM method! Mathematical details see e.g. - D. Arnold et al. Finite Element Exterior Calculus, Homological Techniques, and Applications, Acta Numerica, P. Monk, Finite Element Methods for Maxwell s Equations, Oxford University Press, November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 67
64 2D Discretization of FCC-hh pipe (similar to LHC pipe) FCC-hh design study: 100TeV c.m., 100km circumf. 12mm 18mm The first hadron collider where synchrotron radiation losses play significant role Gmsh triangular mesh Meshing the whole structure is required only for extremely low frequency! Otherwise: Surface Impedance Boundary Condition (SIBC) Design by R. Kersevan, CERN, mesh by T. Egenolf, TU Darmstadt 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 68
65 2D Simulations in the Frequency Domain BeamImpedance2D, PYTHON code using FEniCS finite element package Horizontal Vertical f=100hz f=1mhz 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 69
66 Penetration Depth Surface impedance for coated surface 6.4kHz Vacuum Copper Titanium 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 70
67 Boundary conditions Natural: Neumann (E-field formulation: Magnetic BC) Essential: Dirichlet (E-field formulation: Electric BC) Surface Impedance Boundary Condition (SIBC) Periodic Floquet Open boudary conditions Mur (absorbing BC) Berenger PML (perfectly matched layer) Waveguide ports Solve 2D eigenvalue problem and connect to 3D structure Dedicated boundary conditions for particle beams 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 71
68 The Smith Chart 17 November 2015 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 72
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